Download 2. Subspaces Definition A subset W of a vector space V is called a

2. Subspaces
Definition A subset W of a vector space
V is called a subspace of V if W is itself a
vector space under the addition and scalar
multiplication defined on V .
Theorem
If W is a set of one or more vectors from a
vector space V , then W is a subspace of V
if and only if the following conditions hold.
(a) If u and v are vectors in W , then u +
v is in W .
(b) If k is any scalar and u is any vector in
W , then ku is in W .
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Examples of Subspaces
• A plane through the origin of R3 forms
a subspace of R3.
• A line through the origin of R3 is also
a subspace of R3.
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• Let n be a positive integer, and let Pn(R)
denote the set of all functions expressible in the form
p(x) = a0 + a1x + · · · + anxn
where a0, . . . , an are real numbers. Thus,
Pn(R) consists of the zero function together with all real polynomials of degree n or less. The set Pn(R) is a subspace of the vector space of all realvalued functions as well as being a subspace of P (R).
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• The transpose AT of an m × n matrix
A is the n × m matrix obtained from A
by interchanging rows and columns. A
symmetric matrix is a square matrix A
such that AT = A. The set of all symmetric matrices in Mn×n(F) is a subspace of Mn×n(F).
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• The trace of an n×n matrix A, denoted
tr(A), is the sum of the diagonal entries
of A. The set of n × n matrices having trace equal to zero is a subspace of
Mn×n(F).
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Operations on Vector Spaces
• The addition of two subspaces (of the
same vector space) is defined by: U +
V = {u + v|u ∈ U, v ∈ V }
• The intersection ∩ of two subsets of a
vector space is defined by:
U ∩ V = {w|w ∈ U and w ∈ V }
• A vector space W is called the direct
sum of U and V , denoted U ⊕V , if U and
V are subspaces of W with U ∩ V = {0}
and U + V = W .
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Theorem
Any intersection of subspaces of a vector
space V is also a subspace of V .
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Quiz
True or false?
(a) If V is a vector space and W is a subset
of V that is also a vector space, then
W is a subspace of V .
(b) The empty set is a subspace of every
vector space.
(c) If V is a vector space other than the
zero vector space, then V contains a
subspace W such that W 6= V .
(d) The intersection of any two subsets of
V is a subspace of V .
(e) Any union of subspaces of a vector space
V is a subspace of V .
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